What means a L-cardinal?

Question

What means a L-cardinal? (in the context of the hierarchy of constructibles)

I understand that this is supposed to be a cardinal in the sense of L, but I am not sure to understand what it means.

Answer 1

An $L$-cardinal is just an ordinal $\alpha$ such that $$L\models\mbox{ "$\alpha$ is a cardinal."}$$

Or, perhaps more intuitively, an $L$-cardinal is an ordinal $\alpha$ such that there is no bijection in $L$ from $\alpha$ to any smaller ordinal.

The point is that any time we have a (first-order expressible) property $P$ of ordinals, we can ask about

• The class of ordinals which have property $P$, or

• The class of ordinals which $L$ thinks have property $P$.

In general, these will be different: e.g. perhaps $\alpha$ is an ordinal such that in $V$ (= "reality") there is a bijection between $\alpha$ and $\omega$ but there is no such bijection in $L$; then $\alpha$ is countable but "$L$-uncountable."